377 
1888,] Lord McLaren on an Aplanatic Ohjective. 
the coefficient of a bein^ sec l(nO) or 1 , --- , we observe that 
' cos i{nO) 
as cos (nO) = cos ( - nB) there are two values of r, there are two 
coaxial curves extending in opposite directions, in the manner of 
the common hyperbola. There are also two conjugate polar curves 
having the same asympotes as those of the primary curve. 
4. The inclination of the asymptotes of the primary curves to 
90° 
the axis being, as above, — , their inclination to the axis of the 
conjugate curve is . The index of the conjugate curve 
is accordingly , and its equation is 
n — \ 
By the theory of curves of this form, this is the equation of the 
pedal of the primary curve. But as the angle 0 is reckoned from 
the conjugate axis, the conjugate curve is the pedal of the primary 
turned round through an angle of 90°, a property which is apparently 
peculiar to curves of the prescribed form having fractional indices. 
5. If we consider the equation of the curve, of Equation 8 (above), 
cos 20 = which represents an equilateral hyperbola, we shall 
find that the pole is the centre, and that a is the semi-axis. For, 
by expressing cos 20 in terms of cos 0, we have cos 20= 2 cos^0 - 1. 
Also e = V2, and the equation becomes r‘‘ = ^ ^ | _ 
the equation of the hyperbola from the centre. Accordingly, the 
pole which has been found to be the optical focus in the curves of 
the polar degree is the centre of the quadrilateral system, or point 
of intersection of the asymptotes. 
6. If we regard as a constant in the family of curves of the 
polar degree, we have only one variable parameter, a ; and all curves 
of the same degree are similar, and have their asymptotes inclined at 
the same angle to the axis of symmetry of the curve. But this is 
only an apparent anomaly. The true equation of the curve is 
= cosmB ] but it is only when m = n that the curve possesses 
the optical properties which are here discussed. When m is different 
from 7z, a curve of a prescribed degree may have any inclination of 
asymptotes to axis. 
